In many problems appearing in applied mathematics in the nonlinear ordinary differential systems, as in physics, chemist, economics, etc., if we have a differential system on a manifold of dimension, two of them having a first integral, then its phase portrait is completely determined. While the existence of first integrals for differential systems on manifolds of a dimension higher than two allows to reduce the dimension of the space in as many dimensions as independent first integrals we have. Hence, to know first integrals is important, but the following question appears: Given a differential system, how to know if it has a first integral? The symmetries of many differential systems force the existence of first integrals. This paper has two main objectives. First, we study how to compute first integrals for polynomial differential systems using the so-called Darboux theory of integrability. Furthermore, second, we show how to use the existence of first integrals for finding limit cycles in piecewise differential systems.

中文翻译：

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通过第一积分的可积性和极限环

在应用数学中出现的许多非线性常微分系统问题中，如物理学、化学家、经济学等，如果我们有一个维流形上的微分系统，其中两个有一个第一积分，那么它的相图是完全确定。虽然维数高于二的流形上的微分系统的第一积分的存在允许在与我们拥有的独立第一积分一样多的维度上减少空间的维数。因此，知道一阶积分很重要，但会出现以下问题：给定一个微分系统，如何知道它是否具有一阶积分? 许多微分系统的对称性迫使第一积分的存在。本文有两个主要目标。首先，我们研究如何使用所谓的 Darboux 可积性理论计算多项式微分系统的一阶积分。此外，第二，我们展示了如何使用第一积分的存在来寻找分段微分系统中的极限环。